2 interactive labs · AS & A Level Mathematics 9709
AP and GP: Terms, Sums and Convergence
1.6 Series · Vary a, d or r and n and predict the sum, see why the formulas work and when a GP has a sum to infinity, then test a, b, c with 2b = a + c and b² = ac.
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About this lab
Vary a, d or r and n and predict the sum, see why the formulas work and when a GP has a sum to infinity, then test a, b, c with 2b = a + c and b² = ac.
You will be able to
Calculate the nth term and the sum of the first n terms of an arithmetic progression, using uₙ = a + (n − 1)d and Sₙ = ½n(2a + (n − 1)d) = ½n(a + l).
Calculate the nth term and the sum of the first n terms of a geometric progression, using uₙ = arⁿ⁻¹ and Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.
State the condition |r| < 1 for a geometric progression to converge, calculate S∞ = a/(1 − r), and explain why there is no sum to infinity otherwise.
Use 2b = a + c and b² = ac to decide whether three numbers are in arithmetic or geometric progression, and solve a problem that involves two progressions.
Binomial Expansion: Build It and Find a Coefficient
1.6 Series · Build (a + b)ⁿ one term at a time from ⁿCᵣ, predicting each coefficient, then find the coefficient of a stated power of x or the term independent of x.
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About this lab
Build (a + b)ⁿ one term at a time from ⁿCᵣ, predicting each coefficient, then find the coefficient of a stated power of x or the term independent of x.
You will be able to
Use ⁿCᵣ = n!/(r!(n − r)!), also written (n r), to build the expansion of (a + b)ⁿ for a positive integer n, term by term.
Write the general term T(r + 1) = ⁿCᵣ aⁿ⁻ʳ bʳ with the numbers substituted, keeping each whole bracket, number and sign, inside the power.
Determine the coefficient of a stated power of x, or the term independent of x, by finding r from the power of x, and decide when no such term exists.
